Nonlinear monotone operators with values in \({\mathcal L}(X,Y)\)
let X be a topological vector space, Y an ordered TVS and L(X,Y) the space of continuous linear operators from X into Y. A multivalued operator, \(T:{\mathcal D}(T)\subset X\to Y\), is monotone if \((A_ 1-A_ 2)(x_ 1-x_ 2)\geq 0\) for all \(x_ i\in {\mathcal D}(T)\) and \(A_ i\in T(x_ i)\), \(i=1,2\). In the case \(Y={\mathbb{R}}\), we recover the usual notion of monotone operators. The present situation is motivated and illustrated by the study of subdifferentials of convex functions from X into Y. The main result deals with the upper demicontinuity, upper hemicontinuity and local boundedness of monotone operators from X into Y. Under some restrictions on Y, it is also shown that the subdifferential of a convex operator, \(F:X\to Y\), which has a closed epigraph is a maximum monotone operator.
scientific article; zbMATH DE number 4171776
scientific article